According to this link, you might have to find the location with the help of other NPC's:

After infiltrating the Thalmor Embassy, return to Riverwood and inform Delphine of what you learned from the Thalmor records. Delphine is surprised to know her old colleague and fellow Blade, Esbern, is reportedly still alive. She tasks you with finding the archivist from his hiding place in Riften. Due to the clandestine nature of the Blades, Delphine gives you a secret phrase to deliver to Esbern in order to secure his trust when you locate him. Depending on your standing with the Thieves Guild, the journal will direct you to talk to Brynjolf, or search the Ratway for Esbern. If you are directed to find Brynjolf, you may speak to him, Dirge, or Vekel the Man inside The Ragged Flagon. All three give an option to persuade or bribe them for information on Esbern's location.

Of course, they will reveal that Esbern is in the Ratway Warrens, where his hideout truly is. Of course, what happens after that... You can find out for yourself or meet the spoilers in the link I gave you.

If you fast travel to Riverwood after getting Esbern from the cell, but Esbern is not with you nor at the Sleeping Giant Inn where Delphine is, Delphine will say something like, "What!? You left Esbern alone?" After that, fast travel back to Riften, then back once again to Riverwood. Esbern should then show up at the Sleeping Giant and you will enter while he is in mid-conversation with Delphine. It's definitely a glitch, but the above worked for me

When this happened to me, he was slow walking around in the section of the map near his hideout. Once I went over there, he exited the door then teleported to Riverwood. It’s a little strange, but I was able to go on with the story.

Any source for this? Why does "depositing" an item help?
– Svj0hnSep 14 '15 at 6:21

This also assumes the user has a house. Are you sure it was not just the fast travel that fixed it for you?
– user106385Sep 14 '15 at 6:27

@Svj0hn I was looking on Gamepedia and for some reason depositing an item resets the account like a tick. So when the account resets you'll find that the quest resets and you'll pick up where you left off.
– Young GuiloSep 14 '15 at 6:33

@YoungGuilo Also, what exactly do you mean by "deposit"? Just leave something in a chest? It seems unlikely to reset anything at all.
– Svj0hnSep 14 '15 at 12:58

@Svj0hn Fast Traveling did nothing for me. Esbern still was missing. On the Skyrim forums there was a post to deposit items in your home. I did it and then it worked.
– Young GuiloSep 16 '15 at 6:59